Related papers: Distances from points to planes

Let $d\geq 2$, $A \subset \mathbb{Z}^d$ be finite and not contained in a translate of any hyperplane, and $q \in \mathbb{Z}$ such that $|q| > 1$. We show $$|A+ q \cdot A| \geq (|q|+d+1)|A| - O_{q,d}(1).$$

In 1971 Fedi\u{i} proved the remarkable theorem that the linear second order partial differential operator in the plane with coefficients 1 and f^2 is hypoelliptic provided that f is smooth, vanishes at the origin and is positive otherwise.…

We study the Erd\"os/Falconer distance problem in vector spaces over finite fields. Let ${\Bbb F}_q$ be a finite field with $q$ elements and take $E \subset {\Bbb F}^d_q$, $d \ge 2$. We develop a Fourier analytic machinery, analogous to…

Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for…

In this paper we present several formulae for computing the partial degrees of the defining polynomial of the offset curve to an irreducible affine plane curve given implicitly, and we see how these formulae particularize to the case of…

A plane curve $C\subset\mathbb{P}^2$ of degree $d$ is called \emph{blocking} if every $\mathbb{F}_q$-line in the plane meets $C$ at some $\mathbb{F}_q$-point. We prove that the proportion of blocking curves among those of degree $d$ is…

A difference set is said to have classical parameters if $ (v,k, \lambda) = (\frac{q^d-1}{q-1}, \frac{q^{d-1}-1}{q-1}, \frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian…

Erd\H{o}s asked what is the maximum number $\alpha(n)$ such that every set of $n$ points in the plane with no four on a line contains $\alpha(n)$ points in general position. We consider variants of this question for $d$-dimensional point…

Let $E$ be a proper symmetric subset of $S^{d-1}$, and $C_{\mathbb{F}_q^d}(E)$ be the Cayley graph with the vertex set $\mathbb{F}_q^d$, and two vertices $x$ and $y$ are connected by an edge if $x-y\in E$. Let $k\ge 2$ be a positive…

Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio…

Let $G$ be a $d$-regular graph and let $F\subseteq\{0, 1, 2, \ldots, d\}$ be a list of forbidden out-degrees. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F|<\tfrac{1}{2}d$, then $G$ should admit an $F$-avoiding…

We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the…

We prove that if $A \subset {\Bbb F}_q$ is such that $$|A|>q^{{1/2}+\frac{1}{2d}},$$ then $${\Bbb F}_q^{*} \subset dA^2=A^2+...+A^2 d \text{times},$$ where $$A^2=\{a \cdot a': a,a' \in A\},$$ and where ${\Bbb F}_q^{*}$ denotes the…

We show that if K is a self-similar set in the plane with positive length, then the distance set of K has Hausdorff dimension one.

A set in d dimensional Euclidean space with d larger than 2 having Hausdorff dimension at least d/2 must have distance set with Hausdorff dimension strictly greater than 1/2.

We study criteria attesting that a given graph can not be embedded in the plane so that neighboring vertices are at unit distance apart and the straight line edges do not cross.

Let $\cal{P}$ be an affine invariant property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$…

Let $\theta\in\mathbb{R}^d$. We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $\theta$: the projection of the lattice $\mathbb{Z}^{d+1}$ to the hyperplane of the first $d$ coordinates along the…

A vertex coloring of a graph $G$ is called a $2$-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Recently, Bousquet et al. (Discrete Mathematics, 346(4), 113288, 2023) proved that…

We are interested to bound from below the number of distinct dot products determined by a finite set of points $P$ in the Euclidean plane. In this paper, we build on the work of B. Hanson, O. Roche-Newton, and S. Senger, to obtain the…